Theorem. Suppose we have the conjugation isomorphism defined by (e.g. and are conjugates). Then .

The notion is that we have to show that divides and divides . But why can't we just stop and say divides ? Because, by definition, if divides , wouldn't that imply that ?

April 12th 2010, 07:24 PM

tonio

Quote:

Originally Posted by Sampras

Theorem. Suppose we have the conjugation isomorphism defined by (e.g. and are conjugates). Then .

The notion is that we have to show that divides and divides . But why can't we just stop and say divides ? Because, by definition, if divides , wouldn't that imply that ?

If you already know that the corresponding irreducible polynomials are (monic, of course) and of the same degree, then yes: it is enough to show that one divides the other. Perhaps in this problem though they do not assume this...